A350714 Least positive integer m such that m^12*n = x^4 + y^3 + z^2 for some nonnegative integers x,y,z.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1

Offset

0,8

Comments

4-3-2 Conjecture: a(n) exists for any nonnegative integer n. Equivalently, each nonnegative rational number can be written x^4 + y^3 + z^2 with x,y,z nonnegative rational numbers.

Note that m/n = (m*n^11)/n^12 for any positive integers m and n.

a(n) <= 4 for n <= 40000 with the only exception a(23710) = 5.

a(n) <= 4 for n = 77000..100000, and a(n) = 4 for n = 78367, 79479, 83494, 84694, 85979, 86822, 87395, 87814, 90047, 90278, 92891, 93715.

Qing-Hu Hou verified a(n) <= 4 for 40000 < n < 77000. - Zhi-Wei Sun Feb 04 2022

a(n) <= 5 for 10^5 < n <= 2*10^5, and a(n)=5 for n=107206, 117615, and 148079. - Qing-Hu Hou, Feb 05 2022

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.

Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Example

a(6) = 1 with 1^12*6 = 1^4 + 1^3 + 2^2.
a(7) = 2 with 2^12*7 = 2^4 + 15^3 + 159^2.
a(75) = 4 with 4^12*75 = 122^4 + 1007^3 + 3951^2.
a(1140) = 3 with 3^12*1140 = 0^4 + 531^3 + 21357^2.
a(23710) = 5 with 5^12*23710 = 217^4 + 17897^3 + 232166^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[m=1; Label[bb]; k=m^12; Do[If[SQ[k*n-x^4-y^3], tab=Append[tab, m]; Goto[aa]], {x, 0, (k*n)^(1/4)}, {y, 0, (k*n-x^4)^(1/3)}]; m=m+1; Goto[bb]; Label[aa], {n, 0, 100}]; Print[tab]

Crossrefs

Cf. A000290, A000578, A000583, A280356.

Sequence in context: A329720 A140194 A353282 * A159923 A287957 A003989

Adjacent sequences: A350711 A350712 A350713 * A350715 A350716 A350717

Keyword

nonn

Author

Zhi-Wei Sun, Feb 02 2022

Status

approved